large deviations in and out of equilibrium · 1 1 1 1 1 1 1 l 0 1 conﬁgurations: occupation...

Large deviations in and out of equilibrium

Cécile Appert-Rolland1, Thierry Bodineau2, Bernard Derrida3,Alberto Imparato4, Vivien Lecomte5, Frédéric van Wijland6

Julien Tailleur7, Jorge Kurchan8

1LPT, Orsay 2DMA, Paris 3LPS, Paris 4DPA, Aarhus 5DPMC, Genève & LPMA, Paris6MSC, Paris 7School of Physics, Edinburgh 8ESPCI, Paris

La Herradura - Granada Seminar – 12th September 2010

V. Lecomte (LPMA) Large deviations in and out of eq. 12/09/2010 1 / 1

Introduction Motivations

Motivations

Qρ0 ρ1

Prob(C) ∝ exp{− energy(C)

temperature

}cannot describe

Non-equilibrium steady-state

Prob[ρ(x)]

Equilibrium fluctuations of dynamical observables

Prob[Q]



Motivations

Qρ0 ρ1

profile ρ(x)


temperature

}cannot describe


Prob[ρ(x)]


Prob[Q]



Motivations

time-integrated current Q

ρ0 =ρ ρ1 =ρ


temperature

}cannot describe


Prob[ρ(x)]


Prob[Q]


Exclusion processes System

Exclusion processesmaximal

occupation N

b bb b

bbb

bbbb

γ

α1 1 1

11 1

β

δ1 L

ρ0 ρ1

Configurations: occupation numbers {ni}Exclusion rule: 0 ≤ ni ≤ NMarkov evolution

∂tP({ni}) =∑n′i

[W (n′i → ni)P({n′i})−W (ni → n′i )P({ni})

]Large deviation function of the time-integrated current Q

〈e−sQ〉 ∼ etψ(s) (⇔ determining P(Q))


Exclusion processes Method

Operator representation [Schütz & Sandow PRE 49 2726]maximal

occupation N

b bb b

bbb

bbbb

γ

α1 1 1

11 1

β

δ1 L

ρ0 ρ1

∂tP = WP

W =∑

1≤k≤L−1

[S+

k S−k+1 + S−k S+k+1 − n̂k (1− n̂k+1)− n̂k+1(1− n̂k )

]+ α

[S+

1 − (1− n̂1)]

+ γ[S−1 − n̂1

]+ δ

[S+

L − (1− n̂L)]

+ β[S−L − n̂L

]S± and creation and annihilation operators:

S+|n〉 = (N − n)|n + 1〉 S−|n〉 = n|n − 1〉

S± and Sz = n̂ − N2 are spin operators (with j = N

2 )



Operator representation [Schütz & Sandow PRE 49 2726]maximal

occupation N

b bb b

bbb

bbbb

γ

α1 1 1

11 1

β

δ1 L

ρ0 ρ1

∂tP = WP

W =∑

1≤k≤L−1

[S+


]+ α

[S+

1 − (1− n̂1)]

+ γ[S−1 − n̂1

]+ δ

[S+

L − (1− n̂L)]

+ β[S−L − n̂L

]S± and creation and annihilation operators:

S+|n〉 = (N − n)|n + 1〉 S−|n〉 = n|n − 1〉

S± and Sz = n̂ − N2 are spin operators (with j = N

2 )V. Lecomte (LPMA) Large deviations in and out of eq. 12/09/2010 4 / 1


Large deviation functionmaximal

occupation N

b bb b

bbb

bbbb

γ

α1 1 1

11 1

β

δ1 L

ρ0 ρ1

⟨e−sQ⟩ ∼ etψ(s) with ψ(s) = max Sp W(s)

W(s) =∑

1≤k≤L−1

[S+


]+ α

[S+

1 − (1− n̂1)]

+ γ[S−1 − n̂1

]+ δ

[S+

L es − (1− n̂L)]

+ β[S−L e−s − n̂L

]


Exclusion processes Local rotation

Mapping non-eq to eq [Imparato, VL, van Wijland, arXiv:0911.0564]

Large deviations of the current ψ(s) = max Sp W(s)

W(s) =

invariant by rotation︷︸︸︷∑1≤k≤L−1

~Sk · ~Sk+1

+ α[S+

1 − (1− n̂1)]

+ γ[S−1 − n̂1

]+ δ

[S+

L es − (1− n̂L)]


]

Local transformation

Q−1W(s)Q =∑

1≤k≤L−1

~Sk · ~Sk+1

+ α′[S+

1 − (1− n̂1)]

+ γ′[S−1 − n̂1

]+δ′

[S+

L es′ − (1− n̂L)]

+ β′[S−L e−s′ − n̂L

]︸︷︷︸

describes contact with reservoirs of same densities



Mapping non-eq to eq [Imparato, VL, van Wijland, arXiv:0911.0564]

Large deviations of the current ψ(s) = max Sp W(s)

W(s) =

invariant by rotation︷︸︸︷∑1≤k≤L−1

~Sk · ~Sk+1

+ α[S+

1 − (1− n̂1)]

+ γ[S−1 − n̂1

]+ δ

[S+

L es − (1− n̂L)]


]Local transformation

Q−1W(s)Q =∑

1≤k≤L−1

~Sk · ~Sk+1

+ α′[S+

1 − (1− n̂1)]

+ γ′[S−1 − n̂1

]+δ′

[S+

L es′ − (1− n̂L)]

+ β′[S−L e−s′ − n̂L

]︸︷︷︸

describes contact with reservoirs of same densities



For the current [Imparato, VL, van Wijland, PRE 80 011131]

maximaloccupation

b b bb b b

b

γ

α1 1 1

1 1β

δ1 Lρ0 ρ1

Symmetricexclusionprocess

b b bb b b

b1 1 11 1

1 Lγ′

α′ β′

δ′ρ′ ρ′

System in equilibrium

Localtransformation

Probstationn.non-eq (current)

lProbstationn.

eq (current′)


Exclusion processes Analogy with quantum mechanics

Macroscopic limit [Tailleur, Kurchan, VL, JPA 41 505001]

Qρ0 ρ1

profile ρ(x)

A reminder: propagator in quantum mechanics

〈final|eitH|initial〉

=

∫dz1 . . . dzn〈final|ei∆tH|zn〉〈zn−1|ei∆tH|zn−2〉 . . .

. . . 〈z1|ei∆tH|initial〉

=

∫DpDq exp{i 1

~ S[p,q]︸︷︷︸action

}


Exclusion processes Analogy with quantum mechanics


Qρ0 ρ1

profile ρ(x)

A reminder: propagator in quantum mechanics

〈final|eitH|initial〉 =

∫dz1 . . . dzn〈final|ei∆tH|zn〉〈zn−1|ei∆tH|zn−2〉 . . .

. . . 〈z1|ei∆tH|initial〉

=

∫DpDq exp{i 1

~ S[p,q]︸︷︷︸action

}


Exclusion processes Hydrodynamic limit


Qρ0 ρ1

profile ρ(x)

Using SU(2) coherent states:

〈ρf|etW|ρi〉 =

∫ ρ(t)=ρf

ρ(0)=ρi

DρDρ̂ exp{L S[ρ̂, ρ]︸︷︷︸action

}

〈e−sQ〉 ∼ 〈ρf|etWs |ρi〉 =

∫ ρ(t)=ρf

ρ(0)=ρi

DρDρ̂ exp{L Ss[ρ̂, ρ]︸︷︷︸action

}


Exclusion processes Hydrodynamic limit


Qρ0 ρ1

profile ρ(x)

Same Ss[ρ̂, ρ] as the MSR action of the Langevin evolution:

∂tρ = −∂x[− ∂xρ+ ξ

]〈ξ(x , t)ξ(x ′, t ′)〉 =

1Lρ(1− ρ)︸︷︷︸

density-dependent

δ(x ′ − x)δ(t ′ − t)

One recovers the action of fluctuating hydrodynamics[Spohn, Bertini De Sole Gabrielli Jona-Lasinio Landim]


Exclusion processes result

Result [Imparato, VL, van Wijland, PRE 80 011131]

Large deviation function [Reminder: 〈e−sQ〉 ∼ etψ(s)]

ψ(s) =1Lµ(s)︸︷︷︸

saddle

+D

8L2F(σ′′

2D2µ(s)

)︸︷︷︸

fluctuations

µ(s) = −(arcsinh

√ω)2

ω = (1−es)(e−sρ0−ρ1−(e−s−1)ρ0ρ1

)

Universal function F

F(u) =∑k≥2

(−2u)kB2k−2

Γ(k)Γ(k + 1)

u

F HuL


Exclusion processes result

Result [Imparato, VL, van Wijland, PRE 80 011131]

Large deviation function [Reminder: 〈e−sQ〉 ∼ etψ(s)]

ψ(s) =1Lµ(s)︸︷︷︸

saddle

+D

8L2F(σ′′

2D2µ(s)

)︸︷︷︸

fluctuations

µ(s) = −(arcsinh

√ω)2

ω = (1−es)(e−sρ0−ρ1−(e−s−1)ρ0ρ1

)Universal function F

F(u) =∑k≥2

(−2u)kB2k−2

Γ(k)Γ(k + 1)

u

F HuL


Conclusion

Summary

Approach:operator formalismlarge deviation function

Extensions:fluctuating hydrodynamicssaddle-point method, instantonsintegration of fluctuations (dynamical phase transition)

Open questions:→ Eq↔non-eq mapping in higher dimensions?→ More generic systems of interacting particle?→ Link to other eq↔non-eq mappings?→ Crossover to KPZ? Universal fluctuations?


Supplementary material Large deviation function of the current

Finite-size effects [Appert, Derrida, VL, van Wijland, PRE 78 021122]

For periodic boundary conditions

large deviation function

ψ(s) = L−1ρ(1− ρ)s2︸︷︷︸order 0

+ L−2F(u)︸︷︷︸finite-size

with u = −12ρ(1− ρ)s2

universal function

F(u) =∑k≥2

(−2u)kB2k−2

Γ(k)Γ(k + 1)

scaling of cumulants of the total current Qtot1t 〈Q2

tot〉 ∼ L1t 〈Q2k

tot 〉 ∼ L2k−2 (k ≥ 2)



Finite-size effects [Appert, Derrida, VL, van Wijland, PRE 78 021122]

For periodic boundary conditions

large deviation function

ψ(s) = L−1ρ(1− ρ)s2︸︷︷︸order 0

+ L−2F(u)︸︷︷︸finite-size

with u = −12ρ(1− ρ)s2

universal function

u

F HuL



With a field [Appert, Derrida, VL, van Wijland, PRE 78 021122]

Periodic boundary conditions

For the WASEP:

Driving field E

D = 1, σ = 2ρ(1− ρ)

Large deviation function

ψ(s) = 12s(s − E)

〈Q2〉ct︸︷︷︸

at saddle-point

+ L−2DF(u)︸︷︷︸small fluctuations

(determinant)

with u = −s(s − E)σσ′′

16D2︸︷︷︸can become > 0

u

F HuL

Dynamical phase transitionbetween

stationnary and non-stationnaryprofiles



With a field [Appert, Derrida, VL, van Wijland, PRE 78 021122]

Periodic boundary conditions For the WASEP:Driving field E D = 1, σ = 2ρ(1− ρ)

Large deviation function

ψ(s) = 12s(s − E)

〈Q2〉ct︸︷︷︸

at saddle-point

+ L−2DF(u)︸︷︷︸small fluctuations

(determinant)

with u = −s(s − E)σσ′′

16D2︸︷︷︸can become > 0

u

F HuL

Dynamical phase transitionbetween

stationnary and non-stationnaryprofiles


Mapping non-eq. to eq. for the density profile and the current

For the current [Imparato, VL, van Wijland, PRE 80 011131]

maximaloccupation

b b bb b b

b

γ

α1 1 1

1 1β

δ1 Lρ0 ρ1


b b bb b b

b1 1 11 1

1 Lγ′

α′ β′

δ′ρ′ ρ′


Localtransformation

Probstationn.non-eq (current)

lProbstationn.

eq (current′)


Mapping non-eq. to eq. for the density profile and the current

For the density profile [Tailleur, Kurchan, VL, JPA 41 505001]

maximaloccupation

b b bb b b

b

γ

α1 1 1

1 1β

δ1 Lρ0 ρ1


b b bb b b

b1 1 11 1

1 L


Non-localtransformation

Probstationn.non-eq (profile)

lProbstationn.

eq (profile′)


Mapping non-eq. to eq. for the current

Non-local mapping [Tailleur, Kurchan, VL, JPA 41 505001]

Qρ0 ρ1

profile ρ(x)

Boundary-driven transport model:long-range correlationsbreaking of time-reversal symmetry

Non-local mapping to equilibrium:accounts for long-range correlations(density gradient)non-eq.←→(fixed density)eq.

yields Prob[ρ(x)] through a extremalization principle


large deviations in and out of equilibrium · 1 1 1 1 1 1 1 l 0 1 conﬁgurations: occupation...

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